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Problem № 98415 12-14

A group of psychologists developed a test, after which each person gets a mark, the number $Q$, which is the index of his or her mental abilities $($the greater $Q$, the greater the ability$)$. For the country’s rating, the arithmetic mean of the $Q$ values of all of the inhabitants of this country is taken.

a) A group of citizens of country $A$ emigrated to country $B$. Show that both countries could grow in rating.

b) After that, a group of citizens from country $B$ $($including former ex-migrants from $A\,)$ emigrated to country $A$. Is it possible that the ratings of both countries have grown again?

c) A group of citizens from country $A$ emigrated to country $B$, and group of citizens from country $B$ emigrated to country $C$. As a result, each country’s ratings was higher than the original ones. After that, the direction of migration flows changed to the opposite direction – part of the residents of $C$ moved to $B$, and part of the residents of $B$ migrated to $A$. It turned out that as a result, the ratings of all three countries increased again $($compared to those that were after the first move, but before the second$)$. $($This is, in any case, what the news agencies of these countries say$)$. Can this be so $($if so, how, if not, why$)$?
$($It is assumed that during the considered time, the number of citizens $Q$ did not change, no one died and no one was born$)$.

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